Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities

The authors derive the following multiple solution result for a class of Landesman-Lazer type problems: Consider the boundary value problem (BVP) −Δu−λu=f(x,u)−h(x) in Ω , u=0 on ∂Ω . Here Ω is a bounded open subset of R N (N≥1) with smooth boundary ∂Ω , and f=f(x,s) is Hölder continuous in x , uniformly for s in bounded intervals of R , and locally Lipschitz continuous in s , uniformly for x∈Ω . Let λ 1 be the first eigenvalue of −Δ in Ω with Dirichlet boundary conditions with corresponding eigenfunction φ , φ>0 in Ω , ∫ Ω φ 2 (x)dx=1 . If lim s→+∞ (f(x,s)/s)=0 , i.e. f is sublinear at +∞ , but may have unrestricted growth as s→−∞ , then essentially for any h∈C 0,α (Ω) satisfying ∫ Ω f − (x)φ(x)dx>∫ Ω h(x)φ(x)dx>∫ Ω f + (x)φ(x)dx , where f + (x)=lim sup s→+∞ f(x,s) , f − (x)=lim inf s→+∞ f(x,s) , the problem (BVP) with λ near λ 1 will have at least one solution for λ≤λ 1 and at least two solutions for λ>λ 1 .